We study transformations of automata and games using Muller conditions into equivalent ones using parity or Rabin conditions. We present two transformations, one that turns a deterministic Muller automaton into an equivalent deterministic parity automaton, and another that provides an equivalent history-deterministic Rabin automaton. We show a strong optimality result: the obtained automata are minimal amongst those that can be derived from the original automaton by duplication of states. We introduce the notions of locally bijective morphisms and history-deterministic mappings to formalise the correctness and optimality of these transformations.

The proposed transformations are based on a novel structure, called the alternating cycle decomposition, inspired by and extending Zielonka trees. In addition to providing optimal transformations of automata, the alternating cycle decomposition offers fundamental information on their structure. We use this information to give crisp characterisations on the possibility of relabelling automata with different acceptance conditions and to perform a systematic study of a normal form for parity automata.

This paper introduces a robust class of functions from finite words to integers that we call Z-polyregular functions. We show that it admits natural characterizations in terms of logics, Z-rational expressions, Z-rational series and transducers.

We then study two subclass membership problems. First, we show that the asymptotic growth rate of a function is computable, and corresponds to the minimal number of variables required to represent it using logical formulas. Second, we show that first-order definability of Z-polyregular functions is decidable. To show the latter, we introduce an original notion of residual transducer, and provide a semantic characterization based on aperiodicity.

We introduce the notion of universal graphs as a tool for constructing algorithms solving games of infinite duration such as parity games and mean payoff games. In the first part we develop the theory of universal graphs, with two goals: showing an equivalence and normalisation result between different recently introduced related models, and constructing generic value iteration algorithms for any positionally determined objective. In the second part we give four applications: to parity games, to mean payoff games, to a disjunction between a parity and a mean payoff objective, and to disjunctions of several mean payoff objectives. For each of these four cases we construct algorithms achieving or improving over the best known time and space complexity.

In this work we prove decidability of the model-checking problem for safe recursion schemes against properties defined by alternating B-automata. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes.

Higher-order recursion schemes are an expressive formalism used to define languages of finite and infinite ranked trees by means of fixed points of lambda terms. They extend regular and context-free grammars, and are equivalent in expressive power to the simply typed $\lambda$Y-calculus and collapsible pushdown automata. Safety in a syntactic restriction which limits their expressive power.

The class of alternating B-automata is an extension of alternating parity automata over infinite trees; it enhances them with counting features that can be used to describe boundedness properties.

We show that the existence of a first-order formula separa- ting two monadic second order formulas over countable ordinal words is decidable. This extends the work of Henckell and Almeida on finite words, and of Place and Zeitoun on $\omega$-words. For this, we develop the algebraic concept of monoid (resp. $\omega$-semigroup, resp. ordinal monoid) with aperiodic merge, an extension of monoids (resp. $\omega$-semigroup, resp. ordinal monoid) that explicitly includes a new operation capturing the loss of precision induced by first-order indistinguishability. We also show the computability of FO-pointlike sets, and the decidability of the covering problem for first-order logic on countable ordinal words.

In this paper, we look at good-for-games Rabin automata that recognise a Muller language (a language that is entirely characterised by the set of letters that appear infinitely often in each word). We establish that minimal such automata are exactly of the same size as the minimal memory required for winning Muller games that have this language as their winning condition. We show how to effectively construct such minimal automata. Finally, we establish that these automata can be exponentially more succinct than equivalent deterministic ones, thus proving as a consequence that chromatic memory for winning a Muller game can be exponentially larger than unconstrained memory.

In this paper, we consider infinitely sorted tree algebras recognising regular language of finite trees. We pursue their analysis under the angle of their asymptotic complexity, i.e. the asymptotic size of the sorts as a function of the number of variables involved.

Our main result establishes an equivalence between the languages recognised by algebras of polynomial complexity and the languages that can be described by nominal word automata that parse linearisation of the trees. On the way, we show that for such algebras, having polynomial complexity corresponds to having uniformly boundedly many orbits under permutation of the variables, or having a notion of bounded support (in a sense similar to the one in nominal sets).

We also show that being recognisable by an algebra of polynomial complexity is a decidable property for a regular language of trees.

Given an MSO formula $\phi$ with free variables $x_1,\dots <!--FUNCTION\; APPLICATION-->,x_k$, one can define the function $♯\phi$ mapping a word w to the number of valuations satisfying $\phi$ in $w$. In this paper, we introduce the class of Z-linear combinations of such functions, that we call Z-polyregular functions. Indeed, it turns out to be closely related to the well-studied class of polyregular functions.

The main result of this paper solve two natural decision problems for Z-polyregular functions. First, we show that one can minimise the number $k\ge 0$ of free variables which are needed to describe a function. Then, we show how to decide if a function can be defined using first-order formulas, by extending the notion of residual automaton and providing an original semantic characterisation based on aperiodicity. We also connect this class of functions to Z-rational series.

In this paper, we look at good-for-games Rabin automata that recognise a Muller language (a language that is entirely characterised by the set of letters that appear infinitely often in each word). We establish that minimal such automata are exactly of the same size as the minimal memory required for winning Muller games that have this language as their winning condition. We show how to effectively construct such minimal automata. Finally, we establish that these automata can be exponentially more succinct than equivalent deterministic ones, thus proving as a consequence that chromatic memory for winning a Muller game can be exponentially larger than unconstrained memory.

We show that the existence of a first-order formula separating two monadic second order formulas over countable ordinal words is decidable. This extends the work of Henckell and Almeida on finite words, and of Place and Zeitoun on $\omega$-words. For this, we develop the algebraic concept of monoid (resp. $\omega$-semigroup, resp. ordinal monoid) with aperiodic merge, an extension of monoids (resp. -semigroup, resp. ordinal monoid) that explicitly includes a new operation capturing the loss of precision induced by first-order indistinguishability. We also show the computability of FO-pointlike sets, and the decidability of the covering problem for first-order logic on countable ordinal words.

In this paper, we initiate a study of the expressive power of tree algebras, and more generally infinitely sorted algebras, based on their asymptotic complexity. We provide a characterization of the expressiveness of tree algebras of bounded complexity.

Tree algebras in many of their forms, such as clones, hyperclones, operads, etc, as well as other kind of algebras, are infinitely sorted: the carrier is a multi sorted set indexed by a parameter that can be interpreted as the number of variables or hole types. Finite such algebras—meaning when all sorts are finite—can be classified depending on the asymptotic size of the carrier sets as a function of the parameter, that we call the complexity of the algebra. This naturally defines the notions of algebras of bounded, linear, polynomial, exponential or doubly exponential complexity...

We initiate in this work a program of analysis of the complexity of infinitely sorted algebras. Our main result precisely characterizes the tree algebras of bounded complexity based on the languages that they recognize as Boolean closures of simple languages. Along the way, we prove that such algebras that are syntactic (minimal for a language) are exactly those in which, as soon as there are sufficiently many variables, the elements are invariant under permutation of the variables. Tree algebras in many of their forms, such as clones, hyperclones, operads, etc, as well as other kind of algebras, are infinitely sorted: the carrier is a multi sorted set indexed by a parameter that can be interpreted as the number of variables or hole types. Finite such algebras—meaning when all sorts are finite—can be classified depending on the asymptotic size of the carrier sets as a function of the parameter, that we call the complexity of the algebra. This naturally defines the notions of algebras of bounded, linear, polynomial, exponential or doubly exponential complexity...

We initiate in this work a program of analysis of the complexity of infinitely sorted algebras. Our main result precisely characterizes the tree algebras of bounded complexity based on the languages that they recognize as Boolean closures of simple languages. Along the way, we prove that such algebras that are syntactic (minimal for a language) are exactly those in which, as soon as there are sufficiently many variables, the elements are invariant under permutation of the variables.

We consider the following question: given an automaton or a game with a Muller condition, how can we efficiently construct an equivalent one with a parity condition? There are several examples of such transformations in the literature, including in the determinisation of Büchi automata.

We define a new transformation called the alternating cycle decomposition, inspired and extending Zielonka’s construction. Our transformation operates on transition systems, encompassing both automata and games, and preserves semantic properties through the existence of a locally bijective morphism. We show a strong optimality result: the obtained parity transition system is minimal both in number of states and number of priorities with respect to locally bijective morphisms.

We give two applications: the first is related to the determinisation of Büchi automata, and the second is to give crisp characterisations on the possibility of relabelling automata with different acceptance conditions.

A chapter describing Simon’s theorem on factorisation forest, and some recent developments.

The factorisation forest theorem describes Ramsey-like decomposition of words in a semigroup.

The chapter describes various equivalent statement of the theorem, in particular using the notion of split. Some applications are described, in particular in the theory of semigroups and for matrices in the tropical semiring. Extensions to infinite linear orders, as well as deterministic variants are also covered.

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial setting, and posed as an open problem the stochastic setting, where the agent is represented by a Markov decision process. In this paper, we show that the stochastic control problem is decidable. Our solution makes significant uses of well quasi orders, of the max-flow min-cut theorem, and of the theory of regular cost functions. We introduce an intermediate problem of independence interest called the sequential flow problem and study its complexity.

In this paper, we present a categorical approach to learning automata over words, in the sense of the $L^{*}$-algorithm of Angluin. This yields a new generic $L^{*}$-like algorithm which can be instantiated for learning deterministic automata, automata weighted over fields, as well as subsequential transducers. The generic nature of our algorithm is obtained by adopting an approach in which automata are simply functors from a particular category representing words to a computation category. We establish that the sufficient properties for yielding the existence of minimal automata (that were disclosed in a previous paper), in combination with some additional hypotheses relative to termination, ensure the correctness of our generic algorithm.

We introduce the notion of universal graphs as a tool for constructing algorithms solving games of infinite duration such as parity games and mean payoff games. In the first part we develop the theory of universal graphs, with two goals: showing an equivalence and normalisation result between different recently introduced related models, and constructing generic value iteration algorithms for any positionally determined objective. In the second part we give four applications: to parity games, to mean payoff games, and to combinations of them (in the form of disjunctions of objectives). For each of these four cases we construct algorithms achieving or improving over the best known time and space complexity.

In this paper we regard languages and their acceptors – such as deterministic or weighted automata, transducers, or monoids – as functors from input categories that specify the type of the languages and of the machines to categories that specify the type of outputs. Our results are as follows: a) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. b) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. c) We show how this framework can be instantiated to unify several phenomena in automata theory, starting with determinization, minimization and syntactic algebras. We provide explanations of Choffrut’s minimization algorithm for subsequential transducers and of Brzozowski’s minimization algorithm in this setting.

Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed $\lambda Y$-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes.

In this paper, we present a categorical approach to learning automata over words, in the sense of the $L^{*}$-algorithm of Angluin. This yields a new generic $L^{*}$-like algorithm which can be instantiated for learning deterministic automata, automata weighted over fields, as well as subsequential transducers. The generic nature of our algorithm is obtained by adopting an approach in which automata are simply functors from a particular category representing words to a ”computation category”. We establish that the sufficient properties for yielding the existence of minimal automata (that were disclosed in a previous paper), in combination with some additional hypotheses relative to termination, ensure the correctness of our generic algorithm.

In this paper, we are interested in automata over infinite words and infinite duration games, that we view as general transition systems. We study transformations of systems using a Muller condition into ones using a parity condition, extending Zielonka’s construction. We introduce the alternating cycle decomposition transformation, and we prove a strong optimality result: for any given deterministic Muller automaton, the obtained parity automaton is minimal both in size and number of priorities among those automata admitting a morphism into the original Muller automaton. We give two applications. The first is an improvement in the process of determinisation of Büchi automata into parity automata by Piterman and Schewe. The second is to present characterisations on the possibility of relabelling automata with different acceptance conditions.

Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed $\lambda Y$-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes.

In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that $f\le g$, there exists effectively an unambiguous tropical automaton computing h such that $f\le h\le g$. This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different.

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial setting, and posed as an open problem the stochastic setting, where the agent is represented by a Markov decision process. In this paper, we show that the stochastic control problem is decidable. Our solution makes significant uses of well quasi orders, of the max-flow min-cut theorem, and of the theory of regular cost functions.

We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers.

In this paper, we give a self contained presentation of a recent breakthrough in the theory of infinite duration games: the existence of a quasipolynomial time algorithm for solving parity games. We introduce for this purpose two new notions: good for small games automata and universal graphs.

The first object, good for small games automata, induces a generic algorithm for solving games by reduction to safety games. We show that it is in a strong sense equivalent to the second object, universal graphs, which is a combinatorial notion easier to reason with. Our equivalence result is very generic in that it holds for all existential memoryless winning conditions, not only for parity conditions.

This paper is a contribution to the study of parity games and the recent constructions of three quasipolynomial time algorithms for solving them. We revisit a result of Czerwiński, Daviaud, Fijalkow, Jurdziński, Lazić, and Parys witnessing a quasipolynomial barrier for all three quasipolynomial time algorithms. The argument is that all three algorithms can be understood as constructing a so-called separating automaton, and to give a quasipolynomial lower bond on the size of separating automata. We give an alternative proof of this result. The key innovations of this paper are the notion of universal graphs and the idea of saturation.

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.

We show how recent results concerning quantitative forms of automata help providing refined understanding of the properties of a system (for instance, a program). In particular, combining the size-change abstraction together with results concerning the asymptotic behavior of tropical au- tomata yields extremely fine complexity analysis of some pieces of code.

Regular cost functions offer a toolbox for automatically solving problems of existence of bounds, in a way similar to the theory of regular languages. More precisely, it allows to test the existence of bounds for quantities that can be defined in cost monadic second-order logic (a quantitative variant of monadic second-order logic) with inputs that range over finite words, infinite words, finite trees, and (sometimes) infinite trees.

Though the initial results date from the works of Hashiguchi in the early eighties, it is during the last decade that the theory took its current shape and that many new results and applications have been established.

In this tutorial, two connections linking logic with the theory of regular cost functions will be described. The first connection is a proof of a result of Blumensath, Otto and Weyer stating that it is decidable whether the fixpoint of a monadic second- order formula is reached within a bounded number of iterations over the class of infinite trees. The second connection is how non- standard models (and more precisely non-standard analysis) give rise to a unification of the theory of regular cost functions with the one of regular languages.

In this paper we regard languages and their acceptors – such as deterministic or weighted automata or transducers – as functors from input categories that specify the type of the languages and of the machines to categories that specify the type of outputs.

Our results are as follows: a) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. b) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. c) We show how this framework can be instantiated to unify several phenomena in automata theory, starting with determinization and minimization (which have been previously studied from a coalgebraic and duality theoretic perspective). We also show how subsequential transducers can be seen as functors valued in a Kleisli category and explain Choffrut’s minimization algorithm. We also give an alternative proof of correctness of Brzozowski’s minimization algorithm.

Already in the seventies, strong results illustrating the intimate relationship between category theory and automata theory have been described and are still investigated. In this column, we provide a uniform presentation of the basic concepts that underlie minimization results in automata theory. We then use this knowledge for introducing a new model of automata that is an hybrid of deterministic finite automata and automata weighted over a field. These automata are very natural, and enjoy minimization result by design.

The presentation of this paper is indeed categorical in essence, but it assumes no prior knowledge from the reader. It is also non-conventional in that it is neither algebraic, nor co-algebraic oriented.

In this paper, we adopt of categorical approach to the conception of automata classes enjoying minimisation by design. The main instantiation of this construction is the new class of automata that are hybrid between deterministic and vector space automata.

We introduce perfect half space games, in which the goal of Player 2 is to make the sums of encountered multi-dimensional weights diverge in a direction which is consistent with a chosen sequence of perfect half spaces (chosen dynamically by Player 2). We establish that the bounding games of Jurdzinski et al. (ICALP 2015) can be reduced to perfect half space games, which in turn can be translated to the lexicographic energy games of Colcombet and Niwinski, and are positionally determined in a strong sense (Player 2 can play without knowing the current perfect half space). We finally show how perfect half space games and bounding games can be employed to solve multi-dimensional energy parity games in pseudo-polynomial time when both the numbers of energy dimensions and of priorities are fixed, regardless of whether the initial credit is given as part of the input or existentially quantified. This also yields an optimal 2-EXPTIME complexity with given initial credit, where the best known upper bound was non-elementary.

We define a new class of languages of $\omega$-words, strictly extending $\omega$-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of $\omega$-regular expressions where two new variants of the Kleene star $L^{*}$ are added: $L^B$ and $L^S$. These new exponents are used to say that parts of the input word have bounded size, and that parts of the input can have arbitrarily large sizes, respectively. For instance, the expression ${\left(a^{B}b\right)}^{\omega }$ represents the language of infinite words over the letters $a,b$ where there is a common bound on the number of consecutive letters $a$. The expression ${\left(a^{S}b\right)}^{\omega }$ represents a similar language, but this time the distance between consecutive $b$’s is required to tend toward the infinite. We develop a theory for these languages, with a focus on decidability and closure. We define an equivalent automaton model, extending Büchi automata. The main technical result is a complementation lemma that works for languages where only one type of exponent—either $L^B$ or $L^S$—is used. We use the closure and decidability results to obtain partial decidability results for the logic MSOLB, a logic obtained by extending monadic second-order logic with new quantifiers that speak about the size of sets.

In this paper, we exhibit a one-to-one correspondence between $\omega$-regular languages and a subclass of regular cost functions over finite words, called $\omega$-regular like cost functions. This bridge between the two models allows one to readily import classical results such as the last appearance record or the McNaughton-Safra constructions to the realm of regular cost functions. In combination with game theoretic techniques, this also yields an optimal and simple procedure of history-determinisation for cost automata, a central result in the theory of regular cost functions.

Distance automata are automata weighted over the semiring $(N\cup \{\infty \},\min <!--FUNCTION\; APPLICATION-->,+)$ (the tropical semiring). Such automata compute functions from words to $N\cup \left\{\infty \right\}$. It is known from Krob that the problems of deciding ‘$f\le g$’ or ‘$f=g$’ for $f$ and $g$ computed by distance automata is an undecidable problem. The main contribution of this paper is to show that an approximation of this problem is decidable.

We present an algorithm which, given $\epsilon >0$ and two functions $f,g$ computed by distance automata, answers “yes” if $f\le (1-\epsilon )g$, “no” if $f\nleqq g$, and may answer “yes” or “no” in all other cases. This result highly refines previously known decidability results of the same type.

The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation of the closure under products of sets of matrices over the tropical semiring.

We also establish another theorem, of affine domination, stating that previously known decision procedures for cost-automata have an improved precision when used over distance automata.

We introduce games with (bound) guess actions. These are games in which the players may be asked along the play to provide numbers that need to satisfy some bounding constraints. These are natural extensions of domination games occurring in the regular cost function theory. In this paper we consider more specifically the case where the constraints to be bounded are regular cost functions, and the long term goal is an $\omega$-regular winning condition. We show that such games are decidable on finite arenas.

Regular cost functions form a quantitative extension of regular languages that share the array of characterisations the latter possess. In this theory, functions are treated only up to preservation of boundedness on all subsets of the domain. In this work, we subject the well known distance automata (also called min-automata), and their dual max-automata to this framework, and obtain a number of effective characterisations in terms of logic, expressions and algebra.

We prove that satisfiability over infinite words is decidable for a fragment of asymptotic monadic second-order logic. In this fragment we only allow formulae of the form $\exists <!--FUNCTION\; APPLICATION-->t\forall <!--FUNCTION\; APPLICATION-->s\exists <!--FUNCTION\; APPLICATION-->r\varphi (r,s,t)$, where $\varphi$ does not use quantifiers over number variables, and variables r and s can be only used simul- taneously, in subformulae of the form $s<f\left(x\right)\le r$.

The notion of orbit finite data monoid was recently introduced by Bojanczyk as an algebraic object for defining recognizable languages of data words. Following Buchi’s approach, we introduce a variant of monadic second-order logic with data equality tests that captures precisely the data languages recognizable by orbit finite data monoids. We also establish, following this time the approach of Schutzenberger, McNaughton and Papert, that the first-order fragment of this logic defines exactly the data languages recognizable by aperiodic orbit finite data monoids. Finally, we consider another variant of the logic that can be interpreted over generic structures with data. The data languages defined in this variant are also recognized by unambiguous finite memory automata.

Determinism of devices is a key aspect throughout all of computer science, simply for considerations of efficiency of the implementation. One possible way (among others) to relax this notion is to consider unambiguous machines: nondeterministic machines that have at most one accepting input on each entry.

In this talk, we will investigate the nature of non-ambiguity in automata theory, surveying the cases of standard finite words up to infinite trees, as well as data-words. The goal of this talk will be to show how this notion of unambiguity is so far not well understood, yielding to difficult open questions. In the last part, I will present recent results with Gabriele Puppis and Michal Skrypczak that provide a deep insight in the data-word case.

In this paper, we study several sublogics of monadic second-order logic over countable linear orderings, such as first-order logic, first-order logic on cuts, weak monadic second-order logic, weak monadic second-ordered logic with cuts, as well as fragments of monadic second-order logic in which sets have to be well ordered or scattered.

We give decidable algebraic characterizations of all these logics and compare their respective expressive power.

Given a formula phi(x,X) positive in X , the boundedness problem asks whether the fixpoint induced by phi is reached within some uniform bound independent of the structure (i.e. whether the fixpoint is spurious, and can in fact be captured by a finite unfolding of the formula). In this paper, we study the boundedness problem when phi is in the guarded fragment or guarded negation fragment of first-order logic, or the fixpoint extensions of these logics. It is known that guarded logics have many desirable computational and model theoretic properties, including in some cases decidable boundedness. We prove that boundedness for guarded negation fixpoint logic is decidable, and even 2EXPTIME-complete. Our proof extends the connection between guarded logics and automata, reducing boundedness for guarded logics to a question about cost automata on trees, a type of automaton with counters that assigns a natural number to each input rather than just a boolean.

A new paradigm, called combinatorial expressions, for computing functions expressing properties over an infinite domains is introduced. The main result is a generic technique, for showing indefinability of certain functions by the expressions, which uses a result, namely Hales-Jewett theorem, from Ramsey theory. An application of the technique for proving inexpressibility results for logics on metafinite structures is given. Apart from this, some extensions and normal form theorems are also presented.

We study two fragments of a fixpoint logic on data words that enjoy Boolean closure, and contain previously defined logics, namely two-variable first order logic and DataLTL. Our main contribution is the separation of these two classes using a reduction to circuit inexpressibility.

The notion of orbit finite data monoid was recently introduced by Bojańczyk as an algebraic object for defining recognizable languages of data words. Following Buuchi’s approach, we introduce a variant of monadic second-order logic with data equality tests that captures precisely the data languages recognizable by orbit finite data monoids. We also establish, following this time the approach of Schutzenberger, McNaughton and Papert, that the first-order fragment of this logic defines exactly the data languages recognizable by aperiodic orbit finite data monoids. Finally, we consider another variant of the logic that can be interpreted over generic structures with data. The data languages defined in this variant are also recognized by unambiguous finite memory automata.

Data $\omega$-words are $\omega$-words where each position is additionally labelled by a data value from an infinite alphabet. They can be seen as graphs equipped with two sorts of edges: ‘next position’ and ‘next position with the same data value’. Based on this view, an extension of Data Automata called Generalized Data Automata (GDA) is introduced. While the decidability of emptiness of GDA is open, the decidability for a subclass class called Büchi GDA is shown using Multicounter Automata. Next a natural fixpoint logic is defined on the graphs of data $\omega$-words and it is shown that the $\mu$-fragment as well as the alternation-free fragment is undecidable. But the fragment which is defined by limiting the number of alternations between future and past formulas is shown to be decidable, by first converting the formulas to equivalent alternating Büchi automata and then to Büchi GDA.

We consider two-player games over graphs and give tight bounds on the memory size of strategies ensuring safety conditions. More specifically, we show that the minimal number of memory states of a strategy ensuring a safety condition is given by the size of the maximal antichain of left quotients with respect to language inclusion. This result holds for all safety conditions without any regularity assumptions, and for all (finite or infinite) graphs of finite degree.

We give several applications of this general principle. In particular, we characterize the exact memory requirements for the opponent in generalized reachability games, and we prove the existence of optimal positional strategies in games with counters.

Max-plus automata (over $\mathbb{N}\cup \{-\infty \}$) are finite devices that map input words to non-negative integers or $-\infty$. In this paper we present (a) an algorithm allowing to compute the asymptotic behaviour of max-plus automata, and (b) an application of this technique to the evaluation of the computational time complexity of programs.

(a) Given a max-plus automaton computing a function $f$, we show that the longest word of value at most $N$ for $f$ has length $\Theta \left(N^{\alpha }\right)$, where $\alpha$ is a computable rational number (or is infinite).

(b) The size-change abstraction (SCA) is an a computational model used for abstracting the behaviour of real programs.

Thanks to result (a), we show that the computational time complexity of terminating SCA instances is decidable: the maximal length of any sequence of transitions is exactly of asymptotic order $\Theta \left(N^{\alpha }\right)$, where $\alpha$ is a computable rational number and where $N$ is the maximal value of the variables in the program.

Regular cost functions provide a quantitative extension of regular languages that retains most of their important properties, such as expressive power and decidability, at least over finite and infinite words and over finite trees. Much less is known over infinite trees.

We consider cost functions over infinite trees defined by an extension of weak monadic second-order logic with a new fixed- point-like operator. We show this logic to be decidable, improving previously known decidability results for cost logics over infinite trees. The proof relies on an equivalence with a form of automata with counters called quasi-weak cost automata, as well as results about converting two-way alternating cost automata to one-way alternating cost automata.

In this paper we introduce so-called asymptotic logics, logics that are meant to reason about weights of elements in a model in a way inspired by topology. Our main subject of study is Asymptotic Monadic Second-Order Logic over infinite words. This is a logic talking about $\omega$-words labelled by integers. It contains full monadic second-order logic and can express asymptotic properties of integers labellings.

We also introduce several variants of this logic and investigate their relationship to the logic MSO+U. In particular, we compare their expressive powers by studying the topological complexity of the different models. Finally, we introduce a certain kind of tiling problems that is equivalent to the satisfiability problem of the weak fragment of asymp- totic monadic second-order logic, i.e., the restriction with quantification over finite sets only.

This chapter is devoted to the presentation of the factorisation forest theorem, a deep result due to Simon, which provides advanced Ramsey-like arguments in the context of algebra, automata, and logic. We present several proofs and several variants the result, as well as applications.

The theory of regular cost functions is a quantitative extension to the classical notion of regularity. A cost function associates to each input a non-negative integer value (or infinity), as opposed to languages which only associate to each input the two values ‘inside’ and ‘outside’. This theory is a continuation of the work on distance automata and similar models. These models of automata have been successfully used for solving the star-height problem, the finite power property, the finite substitution problem, the relative inclusion star-height problem and the boundedness problem for monadic-second order logic over words. Our notion of regularity can be – as in the classical theory of regular languages – equivalently defined in terms of automata, expressions, algebraic recognisability, and by a variant of the monadic second-order logic. Those equivalences are strict extensions of the corresponding classical results. The present paper introduces the cost monadic logic, the quantitative extension to the notion of monadic second-order logic we use, and show that some problems of existence of bounds are decidable for this logic. This is achieved by introducing the corresponding algebraic formalism: stabilisation monoids.

Distance automata are automata weighted over the tropical semiring $(\mathbb{N}\cup \{\infty \},min,+)$. Such automata compute functions from words to $(\mathbb{N}\cup \{\infty \}$. It is known that testing $f<g$ is an undecidable problem for $f$ and $g$ computed by distance automata. The main contribution of this paper is to show that an approximation of this problem becomes decidable.

We present an algorithm which given $ϵ>0$ and $f$, $g$ computed by distance automata answers ”yes” if $f<(1-ϵ)g$, ”no” if there is a word w such that $f\left(w\right)>g\left(w\right)$, and may answer ”yes” or ”no” in all other cases. This result highly refines previously known decidability results of the same type.

The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation to the closure under products of sets of matrices over the tropical semiring. We also provide another theorem of affine domination which shows that previously known decision procedures for cost-automata have an improved precision when used over distance automata.

Gurevich and Rabinovich raised the following question: given a property of a set of rational numbers definable in the real line by a monadic second-order formula, is it possible to define it directly in the rational line? In other words, is it true that the presence of reals at the background do not increase the expressive power of monadic second-order logic?

In this paper, we answer positively this question. The proof in itself is a simple application of classical and more recent techniques. This question will guide us in a tour of results and ideas related to monadic theories of linear orderings.

Weakly definable languages of infinite trees are an expressive subclass of regular tree languages definable in terms of weak monadic second-order logic, or equivalently weak alternating automata. Our main result is that given a Büchi automaton, it is decidable whether the language is weakly definable. We also show that given a parity automaton, it is decidable whether the language is recognizable by a nondeterministic co-Büchi automaton.

The decidability proofs build on recent results about cost automata over infinite trees. These automata use counters to define functions from infinite trees to the natural numbers extended with infinity. We reduce to testing whether the functions defined by certain ”quasi-weak” cost automata are bounded by a finite value.

Cost monadic logic extends monadic second-order logic with the ability to measure the cardinal of sets. In particular, it allows to decide problems related to boundedness questions. In this paper, we provide new decidability results allowing the systematic investigation of questions involving “relative boundedness”. The first contribution in this work is to introduce a suitable logic for such questions. The second is to show the decidability of this logic over finite words. The third contribution is the use of non-standard analysis: we advacate that developing the proofs in the axiomatic system of “relative internal set theory” entails a significant simplification of the proofs.

B-automata are automata that can compute functions from words to non-negative integers or infinity. In this paper we give another semantics to the hierarchical variant of these automata. This new semantics is equivalent to the classical one in the terminology of cost functions: functions computed are equivalent up to a polynomial factor. We provide three applications of the technique.

Our first application shows that it is possible to evaluate, reading once the word from left to right, in deterministic logspace, a B-automaton with the new semantics. A similar evaluation would require polynomial space with the original semantics.

Our second theorem shows that for games with hB-quantitative objectives, there are positional determinacy strategies that are uniformly optimal, i.e., optimal from any starting point.

Finally, we introduce a new form of history-determinism that is uniform in the sense that translation strategies are independent from bounds. We show that uniform history-determinism cannot be enforced for usual B-automata, and we disclose new forms of (equivalent) automata that have this property.

This thesis gives a broad picture of the theory of regular cost functions. Regular cost functions are quantitative extensions to regular languages. A cost function is a function from words (or trees. . .) to the non-negative integers or infinity. These functions are considered modulo an equivalence relation that allows some distortion of the exact values, but preserves the existence of bounds.

Regular cost functions form a sub-class that admits many effectively equivalent charac- terisations in terms of logic, automata, algebra and regular expressions. Some decidability results are deduced.

These results extend on the one hand the classical results on regular languages, and on the other hand the works of Hashiguchi, Simon, Leung, and Kirsten concerning distance automata, the tropical semiring and the star-height problem.

This document describes the current knowledge about regular cost functions for finite words, infinite words, finite trees and infinite trees.

We survey in this paper some variants of the notion of determinism, refining the spectrum between non-determinism and determinism. We present unambiguous automata, strongly unambiguous automata, prophetic automata, guidable automata, and history-deterministic automata. We instantiate these various notions for finite words, infinite words, finite trees, infinite trees, data languages, and cost functions. The main results are underlined and some open problems proposed.

Regular cost functions have been introduced recently as an extension of the notion of regular languages with counting capabilities, which retains strong closure, equivalence, and decidability properties. Cost functions are functions ranging over $N\cup \left\{\infty \right\}$, and considered modulo an equivalence which preserves the existence of bounds over each subset of the domain. A central result in the theory of regular cost functions over words is the duality theorem stating that the two dual forms of automata, namely Band S-automata, are equivalent. The only known proof of this result relies on the translation into an equivalent algebraic formalism. In this paper, we describe direct translations from hierarchical B-automata to history-deterministic S-automata and from hierarchical S-automata to history deterministic B-automata. We furthermore describe how to optimize the number of counters of the output automaton, and how to make them hierarchical. This construction is reminiscent of the famous construction of Safra for the determinization of automata over infinite words.

We develop an algebraic model suitable for recognizing languages of words indexed by countable linear orderings. We prove that this notion of recognizability is effectively equivalent to definability in monadic second-order (MSO) logic. This reproves in particular the decidability of MSO logic over the rationals with order. Our proof also implies the first known collapse result for MSO logic over countable linear orderings.

The notion of orbit finite data monoid was recently introduced by Bojanczyk as an algebraic object for defining recognizable languages of data words.

In this paper, following Buchi’s approach, we introduce the new logic ”rigidly guarded MSO” and show that the languages of data words definable in this logic are exactly the languages recognizable by orbit finite data monoids.

We also establish, following this time the approach of Schutzenberger, McNaughton and Papert, that the first-order variant of this logic exactly defines the languages recognizable by aperiodic orbit finite data monoids.

The objective of this survey is to present the ideal theory of monoids, the so-called Green’s relations, and to illustrate the usefulness of this tool for solving automata related questions. We use Green’s relations for proving four classical results related to automata theory: The result of Schutzenberger characterizing star-free languages, the theorem of factorization forests of Simon, the characterization of infinite words of decidable monadic theory due to Semenov, and the result of determinization of automata over infinite words of Mc-Naughton.

Regular cost functions have been introduced recently as an extension to the notion of regular languages with counting capabilities, which retains strong closure, equivalence, and decidability properties. The specificity of cost functions is that exact values are not considered, but only estimated.

In this paper, we study the strict subclass of regular temporal cost functions. In such cost functions, it is only allowed to count the number of occurrences of consecutive events. For this reason, this model intends to measure the length of intervals, i.e., a discrete notion of time. We provide various equivalent representations for functions in this class, using automata, and ‘clock based’ reduction to regular language. We show that the conversions are much simpler to obtain, and much more efficient than in the general case of regular cost functions.

Our second aim in this paper is to use temporal cost function as a test-case for exploring the algebraic nature of regular cost functions. Following the seminal ideas of Schutzenberger, this results in a decidable algebraic characterization of regular temporal cost functions inside the class of regular cost functions.

We develop the theory of regular cost functions over finite trees: a quantitative extension to the notion of regular languages of trees: Cost functions map each input (tree) to a value in omega + 1, and are considered modulo an equivalence relation which forgets about specific values, but preserves boundedness of functions on all subsets of the domain. We introduce nondeterministic and alternating finite tree cost automata for describing cost functions. We show that all these forms of automata are effectively equivalent. We also provide decision procedures for them. Finally, follow- ing Büchi’s seminal idea, we use cost automata for provid- ing decision procedures for cost monadic logic, a quantita- tive extension of monadic second order logic.

The theorem of factorization forests of Imre Simon shows the existence of nested factorizations — à la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond.

We provide two improvements to the standard result. First we improve on all previously known bounds. Second, we extend it to ‘every linear ordering’.

We use this last variant in a simplified proof of the translation of recognisable languages over countable scattered linear orderings to languages accepted by automata.

We introduce the notion of regular cost functions: a quantitative extension to the standard theory of regular languages. We provide equivalent characterisations of this notion by means of automata (extending the nested distance desert automata of Kirsten), of history-deterministic automata (history-determinism is a weakening of the standard notion of determinism, that replaces it in this context), and a suitable notion of recognisability by stabilisation monoids. We also provide closure and decidability results.

In this paper we establish a lower bound $\Omega \left({(1.64n)}^n\right)$ for the problem of translating a Büchi word automaton of size $n$ into a deterministic Rabin word automaton when both the Büchi and the Rabin condition label transitions rather than states. This lower bound exactly matches the known upper bound to this problem, namely $\Omega \left({(1.64n)}^n\right)$. Our result entails a lower bound when the input Büchi automaton has (as it is usual) its Büchi acceptance condition labeling states. Those lower bounds remain when the output deterministic Rabin automaton has its Rabin acceptance condition labeling states. Hence, in the standard setting, our result establishes a lower bound of $\Omega \left({(1.64n)}^n\right)$, while the best known lower bound was $\Omega \left({(0.76n)}^n\right)$. A known upper bound in the standard setting is $o\left({(2.66n)}^n\right)$.

We introduce and develop the notion of regular cost functions: a quantitative extension to the standard theory of regular languages. This work is a continuation of the works on distance automata and similar models. These models of automata have been successfully used for solving the star-height problem, the finite power property, the finite substitution problem, the relative inclusion star-height problem or the boundedness problem for monadic-second order logic over words. Our notion of regularity can be – as in the classical theory of regular languages – equivalently defined in terms of automata, of algebraic recognisability, and by a variant of the monadic second-order logic. Those equivalences are strict extensions of the corresponding classical results. The decidability of the limitedness problem of distance automata, as well as all its known variants can be deduced from our results.

Tree-walking automata are a natural sequential model for recognizing tree languages. It is well known that every tree language recognized by a tree-walking automaton is regular. We show that the converse does not hold.

Given a Rabin tree-language and natural numbers i, j, the language is said to be i,j-feasible if it is accepted by a parity automaton using priorities i,i+1,...,j. The i,j-feasibility induces a hierarchy over the Rabin-tree languages called the Mostowski hierarchy. In this paper we prove that the problem of deciding if a language is i,j-feasible is reducible to the uniform universality problem for distanceparity automata. Distance-parity automata form a new model of automata extending both the nested distance desert automata introduced by Kirsten in his proof of decidability of the star-height problem, and parity automata over infinite trees. Distance-parity automata, instead of accepting a language, attach to each tree a cost in omega+1. The uniform universality problem consists in determining if this cost function is bounded by a finite value.

This chapter is devoted to the presentation of the factorisation forest theorem, a deep result due to Simon, which provides advanced Ramsey-like arguments in the context of algebra, au- tomata, and logic. We present several proofs and several variants the result, as well as applications.

We survey operations on (possibly infinite) relational structures that are compatible with logical theories in the sense that, if we apply the operation to given structures then we can compute the theory of the resulting structure from the theories of the arguments (the logics under consideration for the result and the arguments might differ). Besides general compatibility results for these operations we also present several results on restricted classes of structures, and their use for obtaining classes of infinite structures with decidable theories.

The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond.

We provide two improvements to the standard result. First we improve on all previously known bounds for the standard theorem. Second, we extend it to every ‘complete linear ordering’. We use this variant in a simplified proof of complementation of automata over words of countable scattered domain.

Following the idea developed by I. Simon in his theorem of Ramseyan factorisation forests, we develop a result of ‘deterministic factorisations’. This extra determinism property makes it usable on trees (finite or infinite). We apply our result for proving that, over trees, every monadic interpretation is equivalent to the composition of a first-order interpretation (with access to the ancestor relation) and a monadic marking. Using this remark, we give new characterisations for prefix-recognisable structures and for the Caucal hierarchy. Furthermore, we believe that this approach has other potential applications.

The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show that it is still valid for any word indexed by a linear ordering; and we show that it admits a deterministic variant for words indexed by well-orderings. A byproduct of this work is also an improvement on the known bounds for the original result. We apply the first variant for giving a simplified proof of the closure under complementation of rational sets of words indexed by countable scattered linear orderings. We apply the second variant in the analysis of monadic second-order logic over trees, yielding new results on monadic interpretations over trees. Consequences of it are new caracterisations of prefix-recognizable structures and of the Caucal hierarchy.

We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable first-order theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and tree-automatic structures.

Tree-walking automata are a natural sequential model for recognizing languages of finite trees. Such automata walk around the tree and may decide in the end to accept it. It is shown that deterministic tree-walking automata are weaker than nondeterministic tree-walking automata.

It is well known that games with the parity winning condition admit positional determinacy : the winner has always a positional (memoryless) strategy. This property continues to hold if edges rather than vertices are labeled. We show that in this latter case the converse is also true. That is, a winning condition over arbitrary set of colors admits positional determinacy in all games if and only if it can be reduced to a parity condition with some finite number of priorities.

We consider an extension of $\omega$-regular expressions where two new variants of the Kleene star $L^{*}$ are added: $L^B$ and $L^S$. These exponents act as the standard star, but restrict the number of iterations to be bounded (for $L^B$) or to tend toward infinity (for $L^S$). These expressions can define languages that are not omega-regular.

We develop a theory for these languages. We study the decidability and closure questions. We also define an equivalent automaton model, extending Büchi automata. This culminates with a — partial — complementation result.

We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable first-order theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and tree-automatic structures.

Tree-walking automata are a natural sequential model for recognizing tree languages. Every tree language recognized by a tree-walking automaton is regular. In this paper, we present a tree language which is regular but not recognized by any (nondeterministic) tree-walking automaton. This settles a conjecture of Engelfriet, Hoogeboom and Van Best. Moreover, the separating tree language is definable already in first-order logic over a signature containing the left-son, right-son and ancestor relations.

Tree-walking automata are a natural sequential model for recognizing tree languages. It is shown that deterministic tree-walking automata are weaker than nondeterministic tree-walking automata.

We introduce top-down deterministic transducers with rational lookahead (transducer for short) working on infinite terms. We show that for such a transducer $\tilde{T}$, there exists an MSO-transduction $T$ such that for any graph $G$, $unfold\left(T\right(G\left)\right)=\tilde{T}\left(unfold\right(G\left)\right)$. Reciprocally, we show that if an MSO-transduction $T$ “preserves bisimilarity”, then there is a transducer $\tilde{T}$ such that for any graph $G$, $unfold\left(T\right(G\left)\right)=\tilde{T}\left(unfold\right(G\left)\right)$. According to this, transducers can be seen as a complete method of implementation of MSO-transductions that preserve bisimilarity. One application is for transformations of equational systems.

This work is dedicated to the study of infinite structures and graphs which admit a finite representation, to their geometrical properties as well as decidability properties concerning them. Following Courcelle’s work, we focus our study on their representation as solution of equational systems.

We first introduce deterministic transducers (top-down with lookahead for infinite trees). This tool allows us to deal with infinite equational systems, thus extending some previous results, originally stated for finite equational systems.

We then study the stack based families of structures and concentrate ourselves on prefix recognizable ones. We establish various equivalent representations for them, and study their restriction to bounded tree-width. We finally consider the enrichment of those equational systems by an extra operator of fusion.

We then study structures admitting term-automatic presentations. We show once more that those structures admit various equivalent representations.

Finally, we study the family of graphs definable by ground term rewriting and establish some equivalences of representations. We conclude by a study of the nature of those graphs when restricted to bounded tree or clique width.

According to Barthelman and Blumensath, the following families of infinite graphs are isomorphic: (1) prefix-recognisable graphs, (2) graph solutions of VR equational systems and (3) MS interpretations of regular trees. In this paper, we consider the extension of prefix-recognisable graphs to prefix-recognisable structures and of graphs solutions of VR equational systems to structures solutions of positive quantifier free definable (PQFD) equational systems. We extend Barthelman and Blumensath’s result to structures parameterised by infinite graphs by proving that the following families of structures are equivalent: (1) prefix-recognisable structures restricted by a language accepted by an infinite deterministic automaton, (2) solutions of infinite PQFD equational systems and (3) MS interpretations of the unfoldings of infinite deterministic graphs. Furthermore, we show that the addition of a fuse operator, that merges several vertices together, to PQFD equational systems does not increase their expressive power.

We study the partial algebra of typed terms with an associative commutative and idempotent operator (typed AC-terms). The originality lies in the representation of the typing policy by a graph which has a decidable monadic theory.

In this paper we show on two examples that some results on AC-terms can be raised to the level of typed AC-terms. The examples are the results on rational languages (in particular their closure by complement) and the property reachability problem for ground rewrite systems (equivalently process rewrite systems).

We consider a new class of infinite graphs defined as the smallest solution of equational systems with vertex replacement operators and unsynchronised product. We show that those graphs have an equivalent internal representation as graphs of recognizable ground term rewriting systems. Furthermore, we show that, when restricted to bounded tree-width, those graphs are isomorphic to hyperedge replacement equational graphs. Finally, we prove that on a wider family of graphs — interpretations of trees having a decidable monadic theory — the first order theory with reachability is decidable.

We propose an automatic method to enforce trace properties on programs. The programmer specifies the property separately from the program; a program transformer takes the program and the property and automatically produces another “equivalent” pogram satisfying the property. This separation of concerns makes the program easier to develop and maintain. Our approach is both static and dynamic. It integrates static analyses in order to avoid useless transformations. On the other hand, it never rejects programs but adds dynamic checks when necessary. An important challenge is to make this dynamic enforcement as inexpensive as possible. The most obvious application domain is the enforcement of security policies. In particular, a potential use of the method is the securization of mobile code upon receipt.